$A$ uniform solid cylinder of mass $M$ and radius $R$ rotates about a frictionless horizontal axle. Two similar masses $m$ are suspended with the help of two ropes wrapped around the cylinder. If the masses fall down through a distance $h$,the angular velocity of the cylinder will be

$A$ uniform rod is rotating in a gravity-free region with a certain constant angular velocity $\omega$. The variation of tensile stress $\sigma$ with distance $x$ from the axis of rotation is best represented by which of the following graphs?

$A$ cylinder of mass $M$ and radius $r$ is mounted on a frictionless axle over a well. $A$ rope of negligible mass is wrapped around the solid cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be

$A$ massless string is wrapped around a disc of mass $M$ and radius $R$. Another end is tied to a mass $m$ which is initially at height $h$ from the ground level as shown in the figure. If the mass is released,then its velocity while touching the ground level will be:

$A$ ring has inner radius $R_1$ and outer radius $R_2$. It rolls without slipping with a constant angular velocity $\omega$. What is the ratio of the forces $F_1/F_2$ experienced by two particles located on the inner and outer surfaces of the ring?

The velocity of the centre of mass of a rigid rod with respect to an observer $O$ is $\vec v_{cm} = (2\hat i + 3\hat j) \text{ m/s}$. The rod has an angular velocity about its centre of mass given by $\vec \omega = (3\hat j + 4\hat k) \text{ rad/s}$. Let $A$ be a point on the rod with position vector $\vec r = 2(\hat i + \hat k) \text{ m}$ with respect to the centre of mass. The velocity of the point $A$ with respect to $O$ is: